Drawing random data from a given two dimensional distribution function [closed] - numpy

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I have a theoretical distribution and I want to randomly sample in 2D space for the following distribution :
def p(z,m):
E = { 'ft':0.55, 'alpha': 2.99, 'z0':0.191, 'km':0.089, 'kt':0.25 }
S = { 'ft':0.39, 'alpha': 2.15, 'z0':0.121, 'km':0.093, 'kt':-0.175 }
I={ 'ft':0.06, 'alpha': 1.77, 'z0':0.045, 'km':0.096, 'kt':0.0 }
Evalue=E['ft']*np.exp(-1*E['kt']*(m-20))*z**E['alpha']*np.exp(-1*(z/(E['z0']+E['km']*(m-20)))**E['alpha'])
Svalue=S['ft']*np.exp(-1*S['kt']*(m-20))*z**S['alpha']*np.exp(-1*(z/(S['z0']+S['km']*(m-20)))**S['alpha'])
Ivalue=I['ft']*np.exp(-1*I['kt']*(m-20))*z**I['alpha']*np.exp(-1*(z/(I['z0']+I['km']*(m-20)))**I['alpha'])
value=Evalue+Svalue+Ivalue
return value
Update:
I figured out that inverse transform sampling is the appropriate approach to sample data from a probability distribution.
How could I program this method in python for 2D data or is there any library that I can use?

Answer 1

Take a look at Markov chain Monte Carlo (MCMC) methods. Basically you jump around the space of (z, m) points. From wherever you are, you always accept a jump that increases p(z, m). You accept a jump which decreases p(z, m) with some probability. There is a Python library PyMC to carry out that process.

Answer 2

If you want to randomly sample a value from p(z,m), then a simple way to implement this would be to use the 'random' module in python. I show the idea using numpy's version of random:
import numpy as np
import matplotlib.pyplot as plt
def p(z,m):
E = { 'ft':0.55, 'alpha': 2.99, 'z0':0.191, 'km':0.089, 'kt':0.25 }
S = { 'ft':0.39, 'alpha': 2.15, 'z0':0.121, 'km':0.093, 'kt':-0.175 }
I={ 'ft':0.06, 'alpha': 1.77, 'z0':0.045, 'km':0.096, 'kt':0.0 }
Evalue=E['ft']*np.exp(-1*E['kt']*(m-20))*z**E['alpha']*np.exp(-1*(z/(E['z0']+E['km']*(m-20)))**E['alpha'])
Svalue=S['ft']*np.exp(-1*S['kt']*(m-20))*z**S['alpha']*np.exp(-1*(z/(S['z0']+S['km']*(m-20)))**S['alpha'])
Ivalue=I['ft']*np.exp(-1*I['kt']*(m-20))*z**I['alpha']*np.exp(-1*(z/(I['z0']+I['km']*(m-20)))**I['alpha'])
value=Evalue+Svalue+Ivalue
return value
# Define the number of iterations you want for each variable
num_iter_m = 50
num_iter_z = 50
# I then set rand_m to go from 20 to 30, as your function fails for <20
rand_m = (np.random.random(num_iter_m)*10)+20
# z goes from the range 0 - 1
rand_z = (np.random.random(num_iter_z))
# Note, I am sampling from a uniform distribution for m and z. You can use more complicated functions, i.e., Gaussian/Normal shapes or even user defined.
rand_p = np.zeros((len(rand_z), len(rand_m)))
# Fill a grid with the random p(z,m) values
for i in range(len(rand_z)):
for j in range(len(rand_m)):
rand_p[i][j] = p(rand_z[i], rand_m[j])
# Plot
fig = plt.figure(0)
ax1 = fig.add_subplot(211)
ax1.scatter(rand_z, rand_m)
ax1.set_xlabel("z")
ax1.set_ylabel("m")
ax2 = fig.add_subplot(212)
cf = ax2.contourf(rand_z, rand_m, rand_p)
ax2.set_xlabel("z")
ax2.set_ylabel("m")
colbar = plt.colorbar(cf)
colbar.set_label("p(z,m)")
plt.show()
A specific module to use to it in a more sophisticated way would be, e.g., PyMC (
https://github.com/pymc-devs/pymc) or emcee (http://dan.iel.fm/emcee/current/).
If you wanted to sample z and m weighted by the 2D function p(z,m) this is a little more complicated.